Chain Rule — Practice Problems and Worked Solutions

The formulas you need here

17 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Differentiate using the chain rule.

   $$f(x)=\sin \left(2x\right)$$

   Answer: $2\cos \left(2x\right)$

   Step-by-step solution

   1. $$\text{Chain rule: differentiate the outer, then multiply by the inner’s derivative.}$$
   2. $$f^{\prime }(x)=2\cos \left(2x\right)$$
2. 2.

   Differentiate using the chain rule.

   $$f(x)=\sin \left(3x\right)$$

   Answer: $3\cos \left(3x\right)$

   Step-by-step solution

   1. $$\text{Chain rule: differentiate the outer, then multiply by the inner’s derivative.}$$
   2. $$f^{\prime }(x)=3\cos \left(3x\right)$$
3. 3.

   Differentiate using the chain rule.

   $$f(x)=\sin \left(5x\right)$$

   Answer: $5\cos \left(5x\right)$

   Step-by-step solution

   1. $$\text{Chain rule: differentiate the outer, then multiply by the inner’s derivative.}$$
   2. $$f^{\prime }(x)=5\cos \left(5x\right)$$
4. 4.

   Differentiate using the chain rule.

   $$f(x)={\left(2x+3\right)}^4$$

   Answer: $8{\left(2x+3\right)}^3$

   Step-by-step solution

   1. $$\text{Chain rule: differentiate the outer, then multiply by the inner’s derivative.}$$
   2. $$f^{\prime }(x)=8{\left(2x+3\right)}^3$$
5. 5.

   Differentiate using the chain rule.

   $$f(x)={\left(3x+1\right)}^3$$

   Answer: $9{\left(3x+1\right)}^2$

   Step-by-step solution

   1. $$\text{Chain rule: differentiate the outer, then multiply by the inner’s derivative.}$$
   2. $$f^{\prime }(x)=9{\left(3x+1\right)}^2$$
6. 6.

   Differentiate using the chain rule.

   $$f(x)=e^{2x}$$

   Answer: $2e^{2x}$

   Step-by-step solution

   1. $$\text{Chain rule: differentiate the outer, then multiply by the inner’s derivative.}$$
   2. $$f^{\prime }(x)=2e^{2x}$$
7. 7.

   Differentiate using the chain rule.

   $$f(x)=e^{4x}$$

   Answer: $4e^{4x}$

   Step-by-step solution

   1. $$\text{Chain rule: differentiate the outer, then multiply by the inner’s derivative.}$$
   2. $$f^{\prime }(x)=4e^{4x}$$
8. 8.

   Differentiate using the chain rule.

   $$f(x)=\sin \left(6x\right)$$

   Answer: $6\cos \left(6x\right)$

   Step-by-step solution

   1. $$\text{Chain rule: differentiate the outer, then multiply by the inner’s derivative.}$$
   2. $$f^{\prime }(x)=6\cos \left(6x\right)$$
9. 9.

   Differentiate using the chain rule.

   $$f(x)=\sin \left(7x\right)$$

   Answer: $7\cos \left(7x\right)$

   Step-by-step solution

   1. $$\text{Chain rule: differentiate the outer, then multiply by the inner’s derivative.}$$
   2. $$f^{\prime }(x)=7\cos \left(7x\right)$$
10. 10.

    Differentiate using the chain rule.

    $$f(x)={\left(2x+1\right)}^5$$

    Answer: $10{\left(2x+1\right)}^4$

    Step-by-step solution

    1. $$\text{Chain rule: differentiate the outer, then multiply by the inner’s derivative.}$$
    2. $$f^{\prime }(x)=10{\left(2x+1\right)}^4$$
11. 11.

    Differentiate using the chain rule.

    $$f(x)={\left(4x+3\right)}^2$$

    Answer: $32x+24$

    Step-by-step solution

    1. $$\text{Chain rule: differentiate the outer, then multiply by the inner’s derivative.}$$
    2. $$f^{\prime }(x)=32x+24$$
12. 12.

    Differentiate using the chain rule.

    $$f(x)={\left(5x+2\right)}^3$$

    Answer: $15{\left(5x+2\right)}^2$

    Step-by-step solution

    1. $$\text{Chain rule: differentiate the outer, then multiply by the inner’s derivative.}$$
    2. $$f^{\prime }(x)=15{\left(5x+2\right)}^2$$
13. 13.

    Differentiate using the chain rule.

    $$f(x)={\left(3x+4\right)}^4$$

    Answer: $12{\left(3x+4\right)}^3$

    Step-by-step solution

    1. $$\text{Chain rule: differentiate the outer, then multiply by the inner’s derivative.}$$
    2. $$f^{\prime }(x)=12{\left(3x+4\right)}^3$$
14. 14.

    Differentiate using the chain rule.

    $$f(x)={\left(2x+5\right)}^6$$

    Answer: $12{\left(2x+5\right)}^5$

    Step-by-step solution

    1. $$\text{Chain rule: differentiate the outer, then multiply by the inner’s derivative.}$$
    2. $$f^{\prime }(x)=12{\left(2x+5\right)}^5$$
15. 15.

    Differentiate using the chain rule.

    $$f(x)=e^{3x}$$

    Answer: $3e^{3x}$

    Step-by-step solution

    1. $$\text{Chain rule: differentiate the outer, then multiply by the inner’s derivative.}$$
    2. $$f^{\prime }(x)=3e^{3x}$$
16. 16.

    Differentiate using the chain rule.

    $$f(x)=e^{5x}$$

    Answer: $5e^{5x}$

    Step-by-step solution

    1. $$\text{Chain rule: differentiate the outer, then multiply by the inner’s derivative.}$$
    2. $$f^{\prime }(x)=5e^{5x}$$
17. 17.

    Differentiate using the chain rule.

    $$f(x)=e^{6x}$$

    Answer: $6e^{6x}$

    Step-by-step solution

    1. $$\text{Chain rule: differentiate the outer, then multiply by the inner’s derivative.}$$
    2. $$f^{\prime }(x)=6e^{6x}$$

More from Calculus I

• Evaluating Limits
• One-Sided Limits
• Limits at Infinity
• Continuity
• Derivative from First Principles
• Power Rule
• Trigonometric Derivatives
• Exponential & Log Derivatives
• Product Rule
• Quotient Rule
• Implicit Differentiation
• Related Rates
• Curve Analysis
• Optimization
• Antiderivatives
• Definite Integrals
• Fundamental Theorem of Calculus
• u-Substitution
• Area Between Curves
• Volumes of Revolution